On zeros of reciprocal polynomials
By P. Lakatos. Publicationes mathematicae (Debrecen), 2002
56 citations recorded by the source corpus.
The purpose of this paper is to show that all zeros of the reciprocal polynomial Pm(z) = m ∑ k=0 Akz k (z ∈ C) of degree m ≥ 2 with real coefficients Ak ∈ R (i.e. Am 6= 0 and Ak = Am−k for all k = 0, . . . , [ m 2 ] ) are on the unit circle, provided that the ”coefficient condition” |Am| ≥ m−1 ∑ k=1 |Ak −Am| is satisfied. Moreover, if the ”coefficient condition” holds, then all zeros ej (j = 1, 2, . . . , m) can be arranged such that ∣∣ei 2πj m+1 − ej ∣∣ < π m + 1 (j = 1, . . . , m). If m = 2n + 1 is odd, then −1 = en+1 is always a zero, and all zeros of P2n+1 are single. If m = 2n is even, if the ”coefficient condition” holds with equality and if sgn A2n = sgn (−1)(Ak −A2n) for all k = 1, 2, . . . , n with Ak − A2n 6= 0, then un = un+1 = π, the number −1 = en = en+1 is a double zero of P2n. Otherwise all zeros of P2n are single.
Subfields: Applied Mathematics, Numerical Analysis
View Paper: https://www.semanticscholar.org/paper/b369b13b55ba9f05919b72efaddcb9c66baa9487
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In Applied Mathematics
In Numerical Analysis
Citations in the corpus
- Self-Inversive Polynomials with All Zeros on the Unit Circle (2005, background): “Lakatos [3] has recently proved that a polynomial P(x) = ∑mj=0 A j x j ∈ R[x] satisfying the conditions Am− j = A j for j ≤ m and”
- Palindrome-Polynomials with Roots on the Unit Circle (2004, background): “The fact that the existence of unimodular roots for the palindrome under consideration is equivalent to the existence of zeros for the cosine polynomial in (4) is an immediate consequence of the considerations above, equality (3), and the fact that if x = e then σk(x) = 2 cos kθ if k ≥ 1.”